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Let $S$ be a monad on a category $\mathbf C$. If $\mathbf C$ is cocomplete, and the category of algebras $\mathbf C^S$ admits reflexive coequalisers, then $\mathbf C^S$ is cocomplete. Thus, it is exis …

A locally small category $\mathscr C$ is called petite if, for every functor $F : \mathscr C \to \mathscr D$ with locally small codomain, and for every object $D \in \mathscr D$, the presheaf $\mathsc …

In the one-dimensional setting, $\kappa$-sifted categories are studied in §3 of Adámek–Koubek–Velebil's A duality between infinitary varieties and algebraic theories. However, it is shown there (Theor …

Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-categ …

For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr …

It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A …

The earliest reference I can find to the universal property of the presheaf construction is Proposition 9.1 of André's Categories of Functors and Adjoint Functors (1966). There is an earlier reference …

By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I a …

My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? This is exactly the topic of Paré's paper Morphisms of Colimits: from Paths to Profu …

The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\math …

Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a …

For finitary commutative monads on $\mathbf{Set}$, this has been studied in Faro–Kelly's On the canonical algebraic structure of a category. I have reworded Proposition 11 ibid. below in terms of fini …

Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We …

This construction is well known in enriched category theory, and has been studied in the increased generality of a class of weights (which recovers the finite discrete weights, describing finite (co)p …

I'm not sure I've followed your question exactly, so let me rephrase it to how I've understood it, and you can tell me if I've misunderstood. I shall use different letters to make sure I'm not acciden …