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Firstly, the $W$-weighted limit $\lim^W F$ is defined to be a representation of $\operatorname{Psh}_D(W,C(-,F-))$; the definition you've given isn't even well-typed. There is no difference at all in …

Day showed that, for suitable V, any monoidal structure on a (V-)functor category $[C^{\mathrm{op}}, V]$ is essentially determined by its restriction to the representables as $$ F \otimes G = \int^{A, …

I don't know of any definition involving Kan extensions, but (co)ends can be expressed as (co)limits weighted by a hom functor (see e.g. here), so that for $F \colon C^{op} \times C \to D$ the end $\i …

See this answer to much the same question. I would call this the 'lax' slice category, although it's not so common a notion that everyone would know what you meant, so maybe you should keep the scare …

The answer is yes if $A=R\mbox{-Mod}$ has pullbacks (which I'm pretty sure it does). See Tom Leinster's answer to a similar question.

The answer (to the first question) is yes: reflections are always monadic, and the associated monad is idempotent.

Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements and $\widehat{C} = …

According to the Elephant, and these notes, an object X in a category C is indecomposable if given an isomorphism $X \cong \coprod_i U_i$ there is a unique $i$ such that $X \cong U_i$ and $U_j \cong 0 …

That their Cauchy completions are equivalent.

What you're asking is whether every limit in a functor category $[B,C]$ is a pointwise limit. The answer is yes if C is complete, but not always otherwise. Kelly gives an example in Basic Concepts o …

The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$. The defining universal property is the same as for comma objects, …

Realizability: An Introduction To Its Categorical Side by Jaap van Oosten is all about realizability toposes. There are some lecture notes online by Wesley Phoa entitled An introduction to fibrations …

Here is one way to look at it: if V is a monoidal category and $\mathbf{B} V$ is the corresponding one-object bicategory, then a V-category in the usual sense is the same thing as a lax functor $\math …

Yes -- filtered/directed colimits commute with finite limits. See Mac Lane, Categories for the Working Mathematician, theorem IX.2.1. Edit: Oh, I thought you meant colimits, but it seems you meant …

Prod(L, Set) is equivalent to the category of algebras for a finitary monad on Set and so is complete and cocomplete by e.g. Borceux Vol. 2 prop. 4.3.6.