Does the composition of a monic morphism with a terminal morphism make a monic morphism?

Question

Consider $T$ to be a terminal element, $T \stackrel{f}{\rightarrowtail} A$ be a monic morphism (this can be shown by the terminal property of $T$) and $B\stackrel{\tau_B}{\rightarrow} T$ be the unique terminal morphism from $B$ to $T$. Then does this make $B \stackrel{f \circ \tau_B}{\rightarrow} A$ a monomorphism? Why?

Answer 1

In the category of R-modules, $0$ is a terminal object. The morphism $R \longrightarrow 0 \longrightarrow R$ is not monic.