For each object $T$ of the topos $\ECat$, we have a natural bijection $\Hom{\ECat}{T}{\ObjMono{X}{Y}}\cong \Monos{\Sl{\ECat}{T}}{X\times T}{Y\times T}$ (the second term representing the set of monomorhpisms from $X\times T$ to $Y\times T$ in $\Sl{\ECat}{T}$).

Proof. Via the cartesian adjunction, the arrows $g:T\to Y\Sup{X}$ correspond bijectively to arrows $\hat{g} : T\times X \to Y$; the latter are in bijection with arrows $g\tick = \prn{\hat{g},p} : T\times X \to T\times Y$, where $p$ denotes the projection $T\times X \to T$.

We also show that the condition “$\hat{\phi}$ factors through $=\Sub{X}$” is equivalent to “$g\tick$ is a mono”. But the first amounts to the commutativity of the inner rectangle below:

If the rectangle commutes, the composite $\prn{g\times X\times X}\circ\prn{T\times \Delta}$ factors through $K$ as $k\circ\bar{g}$ and the square $(1)$ is a fiber product (taking account of the fact that $T\times\Delta$ is a mono); likewise, if the fibered product of $k$ and $g\times X\times X$ is a square of type $(1)$, then the inner rectangle is commutative (by associativity of fibered products).

The condition on $(1)$ will turn out to be satisfied if and only if $T\times \Delta$ is the equalizer of the composites $\Con{ev}\circ \prn{T\times\pi\Sub{1}}$ and $\Con{ev}\circ \prn{T\times\pi_2}$ (according to a classical lemma) the composites $\hat{g}\circ \prn{T\times\pi\Sub{1}}$ and $\hat{g}\circ\prn{T\times\pi\Sub{2}}$.

It remains to show that this property is equivalent to the fact that $\prn{\hat{g},p}$ is a mono, which can be verified in terms of sets: if we denote by $\prn{t,x\Sub{1},x\Sub{2}}$ an arbitrary element of $T\times X\times X$, the two properties amount to the condition “$\hat{g}\prn{t,x\Sub{1}} = \hat{g}\prn{t,x\Sub{2}}$ entails $x\Sub{1}=x\Sub{2}$”. ∎

If $T=\ObjTerm{\ECat}$, we obtain a natural bijection

Thus the points of $\ObjMono{X}{Y}$ correspond exactly to the monos from $X$ to $Y$ in $\ECat$. But these do not entirely determine the object $\ObjMono{X}{Y}$.