What equivalence relation is being used to define the category of partial maps? Here's what Awodey says in his Category Theory.

For any category $\mathbf{C}$ with pullbacks, define the category $\mathbf{Par}(\mathbf{C})$ of partial maps in $\mathbf{C}$ as follows: the objects are the same as those of $\mathbf{C}$, but an arrow $f:A\to B$ is a pair $(|f|, U_f)$, where $U_f\rightarrowtail B$ is a subobject and $|f|:U_f\to A$ is a suitable equivalence class of arrows, as indicated in the diagram:
  $\require{AMScd}$
  \begin{CD}
    U_f @>|f|>> B\\
    @VVV\\
    A
\end{CD}
  Composition of $(|f|,U_f):A\to B$ and $(|g|,U_g):B\to C$ is given by taking a pullback and then composing to get $(|g\circ f|,|f|^*(U_g))$, as suggested by the following diagram:
  $\require{AMScd}$
  \begin{CD}
    |f|^*(U_g) @>>> U_g @>>|g|> C\\
    @VVV @VVV\\
    U_f @>>|f|> B\\
    @VVV\\
    A
\end{CD}

What is this "suitable equivalence relation" which Awodey is talking about? I know that a partial map $h:X\to Y$ is a map $h:X'\to Y$ where $X'\subset X$. However, this intuition is leading me to think that $f\sim g$ if $f\iota=g\iota$ where $\iota:U_f\to A$ is the inclusion, but this would require $\text{dom}(f)=A=\text{dom}(g)$ which I don't think is correct. Even if you don't know the answer, any ideas which might spur inspiration in me would be much appreciated.
 A: What Steve Awodey (probably) meant is that a morphism $f : A \to B$ in $\mathbf{Par}(\mathcal{C})$ is an equivalence class of pairs $(|f|, U_f)$, where $U_f$ is a subobject of $A$ and $|f|$ is a morphism $U_f \to B$ in $\mathcal{C}$, subject to the equivalence relation that identifies $(|f|, U_f)$ with $(|g|, U_g)$ if and only if there is an isomorphism $\theta : U_f \overset{\cong}{\longrightarrow} U_g$ such that $|g| \circ \theta = |f|$. That is, the equivalence relation is on the pairs $(U_f, |f|)$ rather than just the morphisms $|f|$.
This is consistent with the definition given in the original paper, Categories of partial maps (Edmund Robinson and Pino Rosolini, 1988).
The reason the equivalence relation is introduced is that composition of morphisms in $\mathbf{Par}(\mathcal{C})$ is defined by pullback, meaning the identity and associativity laws would hold only up to isomorphism if we didn't quotient by this equivalence relation.
It is common practice in category theory to identify subobjects $m : U \hookrightarrow A$ and $n : V \hookrightarrow A$ when there is an isomorphism $\theta : U \to V$ such that $n \circ v = m$. When $\mathcal{C}=\mathbf{Set}$, we go even further and simply identify subobjects $m : U \to A$ of sets with their image $m[U] \subseteq A$, since this equivalence relation identifies injections $m : U \hookrightarrow A$ with the subset inclusion function $m[U] \hookrightarrow A$. This allows us to say that a subobject of a set $A$ in $\mathbf{Set}$ 'is' a subset of $A$.
