When Does a Map Induce an Isomorphism or Homomorphism?

[WWGD]

Hi, everyone:
I keep seeing, mostly in homological algebra, the use of "induced
homomorphs" or "induced isomorphisms". I get the idea of what is
going on, but I have not been able to find a formal result that
rigorously explains this, i.e, under what conditions does a map
induce an isomorphism or a homomorphism?. The only patterns
there seems to be in all these induced maps is that they are all
defined in some quotient space of the domain, i.e, if
we have f:X->Y , then the induced maps f* are , or seem to be,
defined in X/~ , for some equiv. relation ~ (e.g, in homotopy and
homology). Also, maybe obviously, f is a continuous map.

Basically: I would like to know a result that would allow me to
give a yes/no answer to the question : does f:X->Y induce an
isomorphism/homomorphism of some sort?

Thanks.

 

[WWGD]

Just in case someone else is interested, I think I have at least a partial answer,
i.e, 2 cases in which maps are induced:

In some cases, maps are induced when we consider categories associated with
spaces, i.e, if we are given a map f (continuous) , f:X->Y , we consider (f_x,O_x)
and (f_y,O_y) as categories , i.e, f_x,f_y are morphisms, and O_x,O_y are
objects, respectfully (of course, we can only do this in some cases only).

As an example, we would consider homology and homotopy as functors.
Anyway, that is a sketch.