Is the identity our old friend \\(\mathrm{hom}\\)? Or rather, in this case, \\(\mathrm{hom}\_{\mathbf{Bool}}\\)?

Edit: I ask because it has the same typing constraints,

\\[
\mathrm{hom} : X^{op} \times X \to \mathbf{Set} \\\\
\mathrm{hom}\_{\mathbf{Bool}} : X^{op} \times X \to \mathbf{Bool}.
\\]