How does a single slit produce coherent light? I was wondering why some apparatus seem to place a single slit right before a double slit. Then I saw this this question. The answer explained it quite well; but I still don't quite understand how a single slit produces coherent light from a partially coherent source.
For those wondering, the apparatus is in this question

the answer did not mention the effects of the single slit

 A: Single slit can have two effects:

*

*It effectively decreases the angular size of the source. Smaller source size results in better time coherence, because there is smaller spread in distances that light has to travel.

*If adjusted properly, the single slit can improve the source coherence in terms of wavelength spread. For demonstration let's assume that the source emits only wavelengths $\lambda_1$ and $\lambda_2$. Consider two slits as a screen for a single slit. A single slit produces diffraction pattern, which is an overlap of diffraction patterns from $\lambda_1$ and $\lambda_2$. Now you can adjust double slit in such a way that diffraction pattern minima from $\lambda_1$ coincide with maxima from $\lambda_2$. Therefore, you have filtered out $\lambda_1$.

In the end, I think that 1. is what really matters in your case.
Update: as PhysicsDave points out below, point 2. is probably wrong. Even if it's not wrong it requires very specific adjustment of the slit parameters. Point 1. is equivalent to the accepted answer in the question you cite: "That's when the first slit becomes useful as it makes light spatially coherent."
A: TL;DR: temporal vs. spatial coherence.
To add to other answers: there exist time coherence (or temporal coherence) and spatial coherence. While the former results from the presence/lack of correlation between the light/photons emitted at the same point at different moments of time, the latter refers to the fact that the light emitted by different parts of an extended light source might be not correlated.
A single slit or a pinhole filters out most of the light coming from the source, leaving only the light coming from a particular direction, i.e., particular place in the object. This, on the one hand, reduces the overall light intensity, but, on the other hand removes the interference due to the waves coming from from difference point sources. This does not make light coherent in time, but this is usually sufficient for the interference experiments. Note that camera obscura and pinhole camera use the same principle.
A: Coherency of light is improved the more "hurdles" we put the light beam through.   By adding slits or apertures photons that are similar in path (direction, angle, source location) will become selected and others off path will be rejected.
The most coherent light is laser light, the process of stimulated emission makes all the photons go in the same direction but in addition the laser cavity is typically small and precisely engineered which also improves coherency.
The additional single slit helps to select similar photons and makes the visible pattern sharper especially for less coherent sources like the sun. The photon has to navigate the single slit, the double slits and then finally the screen all of which affects coherency.
A: The set up of the first linear slit, in the diagram of   the double slits above, is designed to make a line of point sources of light .
A point source with a single frequency input has to follow the inverse square law.

If far enough away from the double slits the wave from the point source  can be modeled as a plane wave. (note that in the diagram there is no distance given between  the point-line  source and the double slits, it has to be large in order for the  plane wave to be a good approximation of the wave front)

A: After reviewing the answer I cited in the question, the answers, and lectures notes on wave coherence, I have come to the following explanation.
Imagine a wave without spatial coherence, so something like:

What the single slit does is taking only a small section of the incoming wave, which would be approximately linear, thus introducing spatial coherence as the wavefront is now approximately parallel to the slits:

Feel free to point out any errors.
