By the definition of the terminal object, $\{*\}$, in the category Set, $\{*\}$ is trivial when appearing on the codomain of a morphism, as then the morphism is unique. It gives non trivial information when appearing on the domain of morphism, for then $f : \{*\} \to B$ picks out a particular element of $B$. There are now as many morphisms as elements of $B$, and $f$ being monomorphism, picks unambiguously elements of the codomain.
I am interested in the case of an initial object. The initial object, call it $I$ is now trivial when appearing on the domain, as it is unique, but it is non-trivial when it appears on the codomain. My question is this:
How can we understand the meaning of $f:A \to I$?
Is there a general method to understand what "meaning" to assign to categorical definitions?