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$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphic to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product of a category $C$ be a group $G$ acting on it is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

In our case, morphisms of the form $(0,f)$ are covariant functor, while the $(1,f)$ are the contravariant functors !

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriate 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphic to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product $G \ltimes C$ of a category $C$ by a group $G$ acting on $C$, is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

In our case, morphisms of the form $(0,f)$ are covariant functor, while the $(1,f)$ are the contravariant functors !

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriate 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is equivalent to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product of a category $C$ be a group $G$ acting on it is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriaté 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphic to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product of a category $C$ be a group $G$ acting on it is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

In our case, morphisms of the form $(0,f)$ are covariant functor, while the $(1,f)$ are the contravariant functors !

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriate 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is equivalent to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product of a category $C$ be a group $G$ acting on it is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

In can also be described as a Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriaté 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint, and it sends $\mathbf{Cat}$ with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category to $\mathbf{Cat'}$.

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is equivalent to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product of a category $C$ be a group $G$ acting on it is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriaté 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.