When a covariant functor could take an initial object to an initial object?

Question

1. Recall that,for a topological space $X$ ,we can associate an singular chain complex $\{S_q(X),\partial_q \}$,so we get an singular homology group $H_*(X)$ of $X$.Therefore,here is a covariant singular homology functor $H_*:Top \to \{graded \ groups,homomorphisms\}$ .
2. My question out of the following exercise:calculate the singular homology group $ H_*(\phi)$,here $\phi$ is the empty set.it's easy to see $S_q(\phi)=0$,so $H_*(\phi)=0 $,note that trivial group is initial object in the category of graded group.This is to say,the singular homology functor takes an initial object to an initial object.(however, the reduced singular homology of empty set is not zero for dimention $-1$)
3. So it's natural to ask when a covariant functor could take an initial object to an initial object? Any help will be greatly appreciated,thanks!

Answer 1

Initial objects are examples of colimits. So all functors which respect colimits will send initial objects to initial objects. Left adjoint functors respect colimits. So Left adjoint functors will send initial objects to initial objects