Why is bringing two object to their initial temperature an irreversible process? In the lectures, given in here, it is stated that 

We can imagine thermodynamic processes which conserve energy but which
  never occur in nature. For example, if we bring a hot object into
  contact with a cold object, we observe that the hot object cools down
  and the cold object heats up until an equilibrium is reached. The
  transfer of heat goes from the hot object to the cold object. We can
  imagine a system, however, in which the heat is instead transferred
  from the cold object to the hot object, and such a system does not
  violate the first law of thermodynamics. The cold object gets colder
  and the hot object gets hotter, but energy is conserved. [...]
An example of an irreversible process is the problem discussed in the
  second paragraph. A hot object is put in contact with a cold object.
  Eventually, they both achieve the same equilibrium temperature. If we
  then separate the objects they remain at the equilibrium temperature
  and do not naturally return to their original temperatures. The
  process of bringing them to the same temperature is irreversible.

However, I can't understand why bringing those objects to their initial temperature is an irreversible process? I mean why shouldn't it be?
 A: Sorry for my english !
You are talking about the principle principle of thermodynamics. Precisely, this is the Clausius statement :
"It is impossible to construct a device which operates on a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body."
https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Chapter5.html 
This is a principle, and it must be taken as such in the context of classical thermodynamics. Cinetic theory and statistical physics can help to understand it.
In the introduction to the chapter on the second law, in his book of thermodynamics, P. M. Morse analyse your questions on the fact that we can not prove the impossibility of doing something.
I have copied some excerpt from "Thermal Physics", Morse, 1964 (Strong and Italic are mine !)

The laws of thermodynamics have a negative quality which distinguishes them from most other laws of physics, which makes
  direct, positive, experimental proof quite difficult. The first law
  may be phrased as saying that energy cannot be destroyed. This sort
  of negative is much harder to demonstrate than is the positive
  statement that gravitational force varies inversely as the square of
  the distance.......In a sense, we have to draw on the experience
  of all of physics to support the first law......
......The second law of thermodynamics also has this negative
  quality, and the lack of direct verification is even more
  noticeable than with the first law..... One way of phrasing the second
  law is that the spontaneous tendency of a system to go toward
  thermodynamic equilibrium cannot be reversed without at the same time
  changing some organized energy, work, into disorganized energy, heat.
  No single experiment will convince one of the validity of such a negative statement. All we can say is that the theory based on it,
  namely thermodynamics, has been and still is, successful in
  interpreting and predicting all thermal phenomena so far.

A: What they mean is that you will be able to return the bodies to their original states, but not without causing a change in the physical state of something else, like, for example, the states of one or more ideal constant temperature reservoirs.  As an example, if you put the body that was originally higher in temperature into contact with a reservoir at its original higher temperature and the body that was originally lower in temperature into contact with a reservoir at its original lower temperature, then, in the end, both bodies will be at their original temperatures.  However, the states of the two reservoirs will have changed.  Even if you use more than two reservoirs (say at intermediate temperature), in the end the states of the various reservoirs will have changed.
If, instead of initially putting the two bodies into direct contact with one another to let them equilibrate, you kept them separate, and brought each of them into contact with a sequence of reservoirs (different for the high temperature body from the low temperature body) running gradually from their initial temperatures to their final temperatures (at the final equilibrium temperature they would have reached by allowing them to irreversibly equilibrate), this would have been a reversible process.  You could have reversed this change and gotten both bodies back to their original states, as well as all the reservoirs.
ADDENDUM
Suppose the cold body started out at 0 C and its final temperature was 100 C.  Now we wish to consider several alternate processes for raising the temperature of the cold body from 0 C to 100 C, and we want to see how close we can come to doing this reversibly (so that it can be returned to its initial state without significantly changing anything else).
The mass of the body is M and its heat capacity is C.  We have at our disposal an array of 101 ideal constant temperature reservoirs, running in equal increments of 1 C, from 0 C for reservoir #0, to 100 C for reservoir #100.  
PROCESS 1:
Forward path:  Put body into contact with reservoir #100 and let them equilibrate.  Heat transferred to body from reservoir 100 = 100MC
Reverse path:
Put body into contact with reservoir #0 and let them equilibrate.  Heat transferred to body from reservoir 0 = -100MC
The net result of this process for the forward and reverse paths is a transfer of 100MC heat from reservoir #100 to reservoir #0.  So the process is far from being reversible.
PROCESS 2:
Forward path:  Put body into contact with reservoir #50 and let them equilibrate.  Heat transferred to body from reservoir #50 = 50MC.  Then put body into contact with reservoir #100 and let them equilibrate.  Heat transferred to body from reservoir #100 = 50MC.  
Reverse path:
Put body into contact with reservoir #50 and let them equilibrate.  Heat transferred to body from reservoir #50 = -50MC.  Then put body into contact with reservoir #0 and let them equilibrate.  Heat transferred to body from reservoir #0 = -50MC.  
The net result of this process for the forward and reverse paths is a transfer of 50MC heat from reservoir #100 to reservoir #0 (reservoir 50 remains unchanged).  So the process is still pretty irreversible, but less so than Process 1, which results in a transfer of 100MC from reservoir #100 to reservoir #0.
Process 3:
Forward path:  Put body into contact with reservoir #1 and let them equilibrate.  Heat transferred to body from reservoir #1 = MC.  Then put body into contact with reservoir #2 and let them equilibrate.  Heat transferred to body from reservoir #2 = 1MC.  Continue this through the sequence of reservoirs in 1 C increments until the body reaches 100 C.
Reverse path:
Put body into contact with reservoir #99 and let them equilibrate.  Heat transferred to body from reservoir #99 = -1MC.  Then put body into contact with reservoir #98 and let them equilibrate.  Heat transferred to body from reservoir #98 = -1MC.  Continue this through the sequence of reservoirs in 1 C increments until the body reaches 0 C.
The net result of this process for the forward and reverse paths is a transfer of 1MC heat from reservoir #100 to reservoir #0 (all reservoirs in between remain unchanged). So this process is not nearly as irreversible as the other two processes which involved much larger temperature changes with many fewer reservoirs.
Now imagine extending this game plan to an infinite number of reservoirs in vanishingly small temperature changes.  In this limit, the process becomes fully reversible.
