Internal monoids are usually defined in monoidal categories, I assume in an abstract slice category $\mathcal C/A$ the monoidal operation is the categorial direct product, now it is indeed the pullback in $\mathcal C$. If $f,g\in Ob \mathcal C/A$, let's denote their pullback accordingly by $f\times_A g$.
Then an internal monoid is an arrow $m:M\to A$ in $\mathcal C$, together with mappings in $\mathcal C/A$
$$\mu:m\times_A m\to m,\ \eta:1_A\to m$$
such that associativity and unitality holds. So, these are basically $\mu:M\times_A M\to M$ and $\eta:A\to M$ arrows in $\mathcal C$ where $M\times_AM$ is the upper left object of the pullback square of $m,m$. They have to make the arising triangles commute. Associativity is about the equality of the two compositions $M\times_A M\times_A M\to M$ and unitality can be written similarly.