Let $J$ be a set of morphisms in a category$C$. An algebraically $J$-injective object is an object $X\in C$ equipped with the structure of, for every morphism $i:A\to B$ in $J$ and every morphism $f:A\to X$, a specified morphism $g:B\to X$ such that $g \circ i = f$.

Properties

Algebras for a pointed endofunctor

Assuming that $C$ is locally small and cocomplete (and $J$ is a small set), given an object $X$, let $F_J X$ be the following pushout:

$\array{
\coprod_{i:A\to B} (C(A,X) \cdot A) & \to & X \\
\downarrow & & \downarrow \\
\coprod_{i:A\to B} (C(A,X) \cdot B) & \to & F_J X
}$

where $\cdot$ represents the copower of $C$ over Set. Then $F_J$ is a pointed endofunctor of $C$, such that the (pointed) endofunctor algebras of $F_J$ are precisely the algebraically $J$-injective objects.

Monadicity

When $C$ is locally small and cocomplete as before, if the algebraically-free monad on the pointed endofunctor $F_J$ exists, then by definition the algebraically $J$-injective objects are its monad algebras. In particular, they are monadic over $C$.