I am looking for a reference concering the definition of epi-1-morphisms and mono-1-morphisms in an arbitrary bicategory. These concepts should be defined somewhere, but I am not able to find a reference. It is not hard to come up with a definition of mono/epi-1-morphisms, but somehow it's not written down.


Thanks for any answers in advance. 
Lukas



Reference for the definition of epi-1-morphisms in bicategories
A definition similar to the one you give in your comment can be found in paragraph 2.8 of the paper



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> Carboni, Aurelio; Johnson, Scott; Street, Ross; Verity, Dominic. Modulated bicategories. J. Pure Appl. Algebra 94 (1994), no. 3, 229--282.
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A functor $F : A \longrightarrow B$ is said to be **pseudomonic** if it is faithful and if for each pair of objects $a,b$ in $A$ and each isomorphism $g : Fa \longrightarrow Fb$ in $B$, there exists a (necessarily unique and invertible) morphism $f : a \longrightarrow b$ in $A$ such that $Ff = g$.


A morphism $j : A \longrightarrow B$ in a bicategory $\mathcal{B}$ is then said to be **pseudomonic** if for each object $X$ of $\mathcal{B}$, the functor $\mathcal{B}(X,j) : \mathcal{B}(X,A) \longrightarrow \mathcal{B}(X,B)$ is pseudomonic (i.e. if $j$ is "representably pseudomonic''). Dually, $j$ is said to be **pseudoepic** if for each object $X$ of $\mathcal{B}$, the functor $\mathcal{B}(j,X) : \mathcal{B}(B,X) \longrightarrow \mathcal{B}(A,X)$ is pseudomonic.


However, the class of (representably) **fully faithful** morphisms, which also generalise monomorphisms in ordinary categories, is far more prevalent in the literature. Fully faithful functors form the right class of a bicategorical factorisation system (essentially surjective on objects, fully faithful) on the $2$-category $\mathbf{Cat}$, and more generally, (representably) fully faithful morphisms are important in the study of the exactness properties of bicategories of stacks, in which they again form the right class of a bicategorical factorisation system.


Note that a morphism with groupoidal codomain in a bicategory is pseudomonic if and only if it is fully faithful.


