Can product objects in a category be exited?

Question

Suppose we have a category $\textbf{C}$ with objects $A$, $B$, and $A \times B$.

Is it guranteed that for every morphism $f : A \rightarrow X$ there is a morphism $f' : A \times B \rightarrow X$, or equivalently, that there is a morphism $\text{id}'_X : A \times B \rightarrow A$?

Answer 1

The product $A\times B$ is by definition accompanied by projection morphism $\pi_1\colon A\times B\to A$ and $\pi_2\colon A\times B\to B$.

So if there exists a morphism $f\colon A\to X$, we readily find a morphism $f\circ \pi_1\colon A\times B\to X$.