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.