we know that in general case every arrow is not function also in the $Div \mathbb{A}b$ there is example of monic that is not injective.
2026-03-27 18:57:07.1774637827
Is there example of equalizer that is not injective?
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Consider the category with three objects $A$, $B$, $C$ and four non-identity arrows, $f\colon A\to B$, $g_1,g_2\colon B\to C$, and $h\colon A\to C$ such that $g_1\circ f= g_2\circ f=h$. Then $f$ is the equalizer of $g_1$ and $g_2$.
Make this category into a concrete category via the functor $F$ to Set sending $A$ to $2$ and $B$ and $C$ to $1$. Then all the non-identity arrows get mapped to the unique arrow to $1$ from their domains. In particular $F(f)$ is the unique arrow $2\to 1$, which is not injective.