If a narrow slit makes monochromatic light coherent, why is light required to pass through the single slit before the double slits in Young's experiment? Wouldn't the double slits individually make light coherent if narrow enough and then form an interference pattern without a single slit?
A single slit does not exactly make light coherent. It makes light spatially coherent in the axis perpendicular to the slit. Spatially coherent means, in essence, that the light appears to come from a very small region. Light from a star, for example, is spatially coherent but light from the sun has much less spatial coherence because the sun appears much larger (angularly) than the star.
If the light illuminating a double slit comes from an angularly small source (e.g., a laser or any light passed through a pinhole or narrow slit), it will have sufficient spatial coherence to form fringes in the double slit experiment.
You are asking whether without the first single slit, why you will have no interference pattern. First of all, you could have an interference pattern, but it might not be observable, due to it being completely blurred.
The reasons why you need a single slit are:
you need a single slit if you work with Sunlight, or a bulb (etc not spatially coherent, meaning not pointlike). There are cases when you do not need a single slit, that is, if you have a point source (like a laser) inside a dark box
the whole point of the single slit is to create a point source, in phase, from the natural sunlight (or bulb)
if you do not use the single slit, you will have photons coming from all directions, and to create a visible interference pattern, you need the photons to come from mostly one single direction
if you want an interference pattern, that is visible, you need the photons to be in phase (coherent, meaning that the phases describing the mathematical formula of the wavefront are not random). There is in the other answer another requirement, called spatial coherence. This would mean that the lightsource is point-like (like a pinhole).
Now there are double slit experiments, where they shoot a single photon at a time, and they repeat that many times. You get an interference pattern, because you have a point source, that is in phase (coherent), and comes from the same direction.
On this site, there is a debate over light being coherent, and what it means. There are two meanings they use here, and I am explaining them just to be clear when we use these expression:
the photons are in phase, meaning that the phases describing the mathematical formula of the wavefront are not random
spatial coherence, which means a point-like source, like a pinhole, or a laser