Let $X_1$ and $X_2$ be two objects of a category with products and let $p_1: X_1 \times X_2 \to X_1$ and $p_2: X_1 \times X_2 \to X_2$ be the projections. Given two morphisms $f_1: X \to X_1$ and $f_2: X \to X_2$, the product of $f_1$ and $f_2$ is the unique morphism $f:X \to X_1 \times X_2$ such that $p_1 \circ f = p_2 \circ f$.
Question. In a category where this makes sense (like monoids, groups, etc.), is there a specific name for the restriction of $f$ to its image (which is a submonoid, subgroup, etc. of $X_1 \times X_2$)?
P.S. At the moment, I am using the term restricted product, but I wonder if there is a better term.
No. This is a quite special construction to have a particular name, and there is none that would be widely recognized.