each pair of arrows $I \oset{\alpha}{\to} L \oset{m}{\rat} G$ with $m$ mono;

(2)

:   $\C$ has a partial map classifier $(F: \C \to \C, \eta: Id \dotarrow F)$.

Languages
---------

The main theme of the paper is “simultaneous” rewriting of entire sets of objects of a category by means of rewriting a single *abstract* object that represents a collection of structures—the *language* of the abstract object. The simplest example of an abstract structure is a plain object of a category to which we associate the language of objects that can be mapped to it; the formal definition is as follows (see also [@ckn:graph-languages-type-graphs-journal]).

\[def:object-language\] Let $A$ be an object of a category $\C$. Given another object $X$, we write $X\dashrightarrow A$ whenever there exists an arrow from $X$ to $A$. We define the *language*[^2] of $A$, denoted by $\mathcal{L}(A)$, as $\mathcal{L}(A) = \{X \in \C \mid X\dashrightarrow A \}$.

Whenever $X \in \mathcal{L}(A)$ holds, we will say that $X$ is *abstracted by* $A$, and $A$ is called the *abstract object*. In the following we will also need to characterize a class of (co-)matches which are represented by a given (co-)match (which is a mono).

\[def:mono-language\] Let $\phi\colon L\rat A$ be 