Let \(\mathcal A\) and \(\mathcal B\) be monoidal \(1\)-categories and let \(F\colon\mathcal A\rightarrow\mathcal B\) be a monoidal functor. The centralizer \(Z(F)\) of \(F\) is the following category:
Its objects are pairs \((b,\gamma)\) of an object \(b\in\mathcal B\) and a natural isomorphism \[\gamma=\gamma_-\colon b\boxtimes F(-)\rightarrow F(-)\boxtimes b\] of functors from \(\mathcal A\) to \(\mathcal B\), called half-braiding, which satisfies the following two compatibility conditions with respect to the monoidal structure of \(\mathcal A\). Firstly, for any \(a_1,a_2\in\mathcal A\), the isomorphism \(\gamma_{a_1 \boxtimes a_2} \colon b \boxtimes F(a_1 \boxtimes a_2) \rightarrow F(a_1 \boxtimes a_2) \boxtimes b\) equals the composite \[b\boxtimes F(a_1\boxtimes a_2)\xrightarrow{\simeq} b\boxtimes F(a_1)\boxtimes F(a_2)\xrightarrow{(\mathrm{id}\boxtimes \gamma_{a_2})\circ (\gamma_{a_1}\boxtimes\mathrm{id})} F(a_1)\boxtimes F(a_2)\boxtimes b\xrightarrow{\simeq} F(a_1\boxtimes a_2) \boxtimes b,\] and secondly \[\gamma_{1_{\mathcal A}} = \left(b\boxtimes F(1_{\mathcal A})\xrightarrow{\simeq} b\boxtimes 1_{\mathcal B}=1_{\mathcal B}\boxtimes b\xrightarrow{\simeq}F(1_{\mathcal A})\boxtimes b\right).\]
Its morphisms are morphisms in \(\mathcal B\) that are compatible with the half-braidings as follows: \[\mathrm{Hom}_{Z(F)}((b,\gamma),(b',\gamma'))=\{f\in\mathrm{Hom}_{\mathcal B}(b,b')\mid \gamma'_a\circ (f\boxtimes\mathrm{id})=(\mathrm{id}\boxtimes f)\circ\gamma_a \colon b\boxtimes F(a)\rightarrow F(a)\boxtimes b\}.\]
The composition is inherited from \(\mathcal B\).