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I am trying to prove Lemma 2.1.1 of Kashiwara-Schapira "Categories and Sheaves":

Let $\beta:I^{\text{op}}\to\mathbf{Set}$ be a functor ($I$ a small category) and let $X\in\mathbf{Set}$. There is a natural isomorphism $$\text{Hom}_{\mathbf{Set}}(X,\lim_{\leftarrow} \beta)\cong \lim_{\leftarrow}\text{Hom}_{\mathbf{Set}}(X,\beta)$$ where $\text{Hom}_{\mathbf{Set}}(X,\beta)$ denotes the functor $I^{\text{op}}\to\mathbf{Set}$ , $i\mapsto \text{Hom}_{\mathbf{Set}}(X,\beta(i))$.

This comes just after they have proved that $$\lim_{\leftarrow} \beta\cong \lbrace \lbrace x_i\rbrace_i \in\prod_i \beta(i)\;|\;\beta(s)(x_j)=x_i\;\text{for all}\; s:i\to j\;\text{in}\;I\rbrace$$ and they say that the lemma "is obvious". Why is it obvious?

(By the way, why are they using the notation of a projective limit? They are defining a general limit in the categorical sense, right?)

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Terminological: this is not a limit in a presheaf category, this is a limit of a presheaf in the category $\mathbf{Set}$. –  Oskar Feb 17 at 22:05
    
"Projective limit" is their terminology for what is normally called a "limit". Their "inductive limit" is what is normally called a "colimit". –  Uday Reddy Feb 18 at 18:56

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Remember the universal property of a limit: the maps from $X$ to $\varprojlim \beta$ are in one-to-one correspondence with the cones over $\beta$ with tip $X$. One builds such cones out of maps from $X$ to $\beta(i)$ which commute with the images of morphisms in $I$ under $\beta$. More precisely, such a cone is specified as an element of $\prod_I \hom(X,\beta(i))$ satisfying the same relations as in the explicit definition you give of a limit. This is just a useful and simple rephrasing of the definition of limit, which can easily get cloaked in technicalities.

The projective limit notation is somewhat standard for an arbitrary limit: projective limits are special cases of limits, and inductive limits of colimits, so it's natural to write $\varprojlim$ for a limit and $\varinjlim$ for a colimit. The alternative is to write $\lim$ and $\text{colim}$. Kashiwara-Schapira is generally considered to be pretty difficult reading, incidentally-if you're new to category theory you might spend some time with a more elementary text alongside.

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Yes, but they haven't defined the limit via a universal property. They have just defined it by setting $$\lim_{\leftarrow} \beta=\text{Hom}_{[I^{\text{op}},\mathbf{Set}]}(1,\beta)$$ –  triwer23 Feb 17 at 21:24
    
OK, then a map from $X$ to $\varprojlim \beta$ is a choice for each $x\in X$ of a natural transformation from $1$ to $\beta,$ i.e. a point in $\beta(i)$ for every $i$ commuting with maps in $I$. This gives a function from $X$ to $\beta(i)$ for every $i$ commuting with maps in $I$, which is what a cone is. –  Kevin Carlson Feb 17 at 21:29

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