I think, that under some mild conditions, there is a more natural and explicit construction (I am writing this off the top of my head, you have to carefully check my statements). The idea is that "adjoining a right adjoint to a morphism, makes the morphism a relation".

First, let us consider an easier (1-dimensional) case, when $\mathbb{C}$ is 2-discrete. If $\mathbb{C}$ is regular, then there is a faithful functor: $$\mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ that embeds $\mathbb{C}$ into its (2,1)-category of internal relations $\mathit{Rel}(\mathbb{C})$ and has the property that the image of every morphism has a right adjoint (in fact the converse is also true --- if a morphism in $\mathit{Rel}(\mathbb{C})$ has a right adjoint, then it comes from $\mathbb{C}$).

An internal relation in $\mathbb{C}$ is a span of morphisms $A \leftarrow R \rightarrow B$, where the legs are jointly monic. A 2-dimensional analogue of an internal relation is a codiscrete cofibred cospan.

So, let us assume, that $\mathbb{C}$ is a cofibrational 2-category. Then one may consider the 2-category $\mathit{Mod}(\mathbb{C})$ of codiscrete cofibred cospans in $\mathbb{C}$ with the (co)fibrational composition (i.e. the dual of discrete fibred spans in $\mathbb{C}^{op}$). There is an embedding: $$\mathbb{C} \rightarrow \mathit{Mod}(\mathbb{C})$$ sending a morphism $f \colon A \rightarrow B$ to the cocomma cospan (i.e. collage) over $A \overset{id}\leftarrow A \overset{f}\rightarrow B$ which has a right adjoint in $\mathit{Mod}(\mathbb{C})$.

(If $\mathbb{C}$ is fibrational, one may also consider the dual construction --- the embedding of $\mathbb{C}$ into the 2-category $\mathit{DFib}(\mathbb{C})$ of discrete fibred spans in $\mathbb{C}$.)

I think, that under some mild conditions, there is a more natural and explicit construction (I am writing this off the top of my head, you have to carefully check my statements). The idea is that "adjoining a right adjoint to a morphism, makes the morphism a relation".

First, let us consider an easier (1-dimensional) case, when $\mathbb{C}$ is 2-discrete. If $\mathbb{C}$ is regular, then there is a faithful functor: $$\mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ that embeds $\mathbb{C}$ into its 2-posetal category of internal relations $\mathit{Rel}(\mathbb{C})$ and has the property that the image of every morphism has a right adjoint (in fact the converse is also true --- if a morphism in $\mathit{Rel}(\mathbb{C})$ has a right adjoint, then it comes from $\mathbb{C}$).

An internal relation in $\mathbb{C}$ is a span of morphisms $A \leftarrow R \rightarrow B$, where the legs are jointly monic. If we take a pushout of such a span, then we obtain a cospan representation of a relation. A 2-dimensional analogue of an internal relation is a codiscrete cofibred cospan.

So, let us assume, that $\mathbb{C}$ is a cofibrational 2-category. Then one may consider the 2-category $\mathit{Mod}(\mathbb{C})$ of codiscrete cofibred cospans in $\mathbb{C}$ with the (co)fibrational composition (i.e. the dual of discrete fibred spans in $\mathbb{C}^{op}$). There is an embedding: $$\mathbb{C} \rightarrow \mathit{Mod}(\mathbb{C})$$ sending a morphism $f \colon A \rightarrow B$ to the cocomma cospan (i.e. collage) over $A \overset{id}\leftarrow A \overset{f}\rightarrow B$ which has a right adjoint in $\mathit{Mod}(\mathbb{C})$.

(If $\mathbb{C}$ is fibrational, one may also consider the dual construction --- the embedding of $\mathbb{C}$ into the 2-category $\mathit{DFib}(\mathbb{C})$ of discrete fibred spans in $\mathbb{C}$.)

I think, that under some mild conditions, there is a more natural and explicit construction (I am writing this off the top of my head, you have to carefully check my statements). The idea is that "adjoining a right adjoint to a morphism, makes the morphism a relation".

First, let us consider an easier (1-dimensional) case, when $\mathbb{C}$ is 2-discrete. If $\mathbb{C}$ is regular, then there is a faithful functor: $$\mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ that embeds $\mathbb{C}$ into its (2,1)-category of internal relations $\mathit{Rel}(\mathbb{C})$ and has the property that the image of every morphism has a right adjoint (in fact the converse is also true --- if a morphism in $\mathit{Rel}(\mathbb{C})$ has a right adjoint, then it comes from $\mathbb{C}$).

An internal relation in $\mathbb{C}$ is a span of morphisms $A \leftarrow R \rightarrow B$, where the legs are jointly monic. A 2-dimensional analogue of an internal relation is a codiscrete cofibred cospan.

So, let us assume, that $\mathbb{C}$ is a cofibrational 2-category. Then one may consider the 2-category $\mathit{Mod}(\mathbb{C})$ of codiscrete cofibred cospans in $\mathbb{C}$ with the (co)fibrational composition. There is an embedding: $$\mathbb{C} \rightarrow \mathit{Mod}(\mathbb{C})$$ sending a morphism $f \colon A \rightarrow B$ to the cocomma cospan (i.e. collage) over $A \overset{id}\leftarrow A \overset{f}\rightarrow B$ which has a right adjoint in $\mathit{Mod}(\mathbb{C})$.

(If $\mathbb{C}$ is fibrational, one may also consider the dual construction --- the embedding of $\mathbb{C}$ into the 2-category $\mathit{DFib}(\mathbb{C})$ of discrete fibred spans in $\mathbb{C}$.)

I think, that under some mild conditions, there is a more natural and explicit construction (I am writing this off the top of my head, you have to carefully check my statements). The idea is that "adjoining a right adjoint to a morphism, makes the morphism a relation".

First, let us consider an easier (1-dimensional) case, when $\mathbb{C}$ is 2-discrete. If $\mathbb{C}$ is regular, then there is a faithful functor: $$\mathbb{C} \rightarrow \mathit{Rel}(\mathbb{C})$$ that embeds $\mathbb{C}$ into its (2,1)-category of internal relations $\mathit{Rel}(\mathbb{C})$ and has the property that the image of every morphism has a right adjoint (in fact the converse is also true --- if a morphism in $\mathit{Rel}(\mathbb{C})$ has a right adjoint, then it comes from $\mathbb{C}$).

An internal relation in $\mathbb{C}$ is a span of morphisms $A \leftarrow R \rightarrow B$, where the legs are jointly monic. A 2-dimensional analogue of an internal relation is a codiscrete cofibred cospan.

So, let us assume, that $\mathbb{C}$ is a cofibrational 2-category. Then one may consider the 2-category $\mathit{Mod}(\mathbb{C})$ of codiscrete cofibred cospans in $\mathbb{C}$ with the (co)fibrational composition (i.e. the dual of discrete fibred spans in $\mathbb{C}^{op}$). There is an embedding: $$\mathbb{C} \rightarrow \mathit{Mod}(\mathbb{C})$$ sending a morphism $f \colon A \rightarrow B$ to the cocomma cospan (i.e. collage) over $A \overset{id}\leftarrow A \overset{f}\rightarrow B$ which has a right adjoint in $\mathit{Mod}(\mathbb{C})$.

(If $\mathbb{C}$ is fibrational, one may also consider the dual construction --- the embedding of $\mathbb{C}$ into the 2-category $\mathit{DFib}(\mathbb{C})$ of discrete fibred spans in $\mathbb{C}$.)