Background and notations: Let's say objects $O_1, O_2$ of some category $C$ are isomorphic (with respect to fixed isomorphisms) to some object $O$, and let's write via an abuse of notation $f$ for both an arbitrary endomorphism $O \to O$ and for the endomorphisms $O_1 \to O_1$, $O_2 \to O_2$ identified with it by the fixed isomorphisms mentioned above.
Then given the Cartesian product $O_1 \times O_2$ in $C$ (let's assume $C$ has arbitrary products), for every pair of endomorphisms $f_1: O_1 \to O_1$, $f_2: O_2 \to O_2$, we get a unique endomorphism $(f_1, f_2): O_1 \times O_2 \to O_1 \times O_2$. Moreover, there is a "mapping" (can we write it as a functor $C \times C \to C$?) sending $(f_1, f_2)$ to the endomorphism $f_2 \circ f_1: O\to O$ of $O$?
Question: Is there a standard / common terminology for this mapping of $(f_1, f_2) \to f_2 \circ f_1$?
A special case of the above would be when we have small categories $C_1$, $C_2$ (considered as objects in the category of small categories, with morphisms being functors), both being isomorphic to some category $C$, and then we can map (via a natural transformation?) the endofunctors $(F_1, F_2)$ of the product category $C_1 \times C_2$ to endofunctors $F_2 \circ F_1$ of $C$.
This is more strongly motivated in the case where $C_1$, $C_2$ are commutative monoids (considered as one-object categories) that are isomorphic to a commutative monoid $C$. Then endomorphisms $f$ of the single object in this category can be uniquely identified with the equal endofunctors $g \mapsto f \circ g$ and $g \mapsto g \circ f$, and given endomorphisms / endofunctors $(f_1, f_2) : C_1 \times C_2 \to C_1 \times C_2$, we get a unique way to map them to endomorphisms / endofunctors $C \to C$, because in this case $f_1 \circ f_2 = f_2 \circ f_1$. So the context above is intended as the most straightforward generalization of this case.
If the monoid operation is written as multiplication, then I guess the special case above is basically just "multiplication up to isomorphism"? And if the monoid operation is written as addition (e.g. in the case of the positive cone of a partially ordered abelian group), then I guess the special case above is basically just "addition up to isomorphism"?
In any case, what I described above seems like a really basic idea amenable to a lot of possible generalizations, so I would be very surprised if it or one of its generalizations does not already have a standard name.
Note: This question seems to be possibly related, or at least possibly asking about a generalization of the case where $O_1, O_2$ are objects in the category of small categories. What is the map $\mathrm{Nat}(F_1,F_2)\times\mathrm{Nat}(G_1,G_2)\to\mathrm{Nat}(F_1\circ G_1,F_2\circ G_2)$? (I'm not sure.) But the accepted answer to that question would not answer this question, which specifically asks for a name for the map.