presheaf

A pre-sheaf is a Contravariant Hom Functor

Category of presheaves

$$\begin{array}{r}obj(PSh(\mathcal{C}))={\mathcal{C}}^{\mathrm{op}}\Rightarrow \mathrm{Set}\\ PSh(\mathcal{C})[F,G]={\int }_{X:\mathcal{C}}FX\to GX\end{array}$$

Let $\mathcal{C}$ be a bi-cartesian closed category Given $F,G:{\mathcal{C}}^{\mathrm{op}}\Rightarrow \mathrm{Set}$

\begin{aligned} (F{\texttimes} G)_0 X=FX\to GX\\ (F+G)_0 X = FX+GX\\ (F\to G)_0 X =FX\to GX * \end{aligned} $$ \* doesn't work

\begin{aligned} PSh_c(H{\texttimes} F, G)\cong PSh_c(H,G^F)\ \int_{X:\mathcal{C} }\mathcal{C}(Y,X){\texttimes} FY\to GY \end{aligned}

$$[[YonedaLemma]]:$$

\left(\int _{Y:C} C(Y,X)\to FY\right) \cong FX

$$Defineexpas:$$

\int_{X:\mathcal{C} }\mathcal{C}(Y,X){\texttimes} FY\to GY

### $(A+B){\texttimes}C\cong A{\texttimes}B+B{\texttimes}C$

\begin{aligned} &\mathcal{C} ((A+B){\texttimes} C,X)\ \cong~ &\mathcal{C} (A+B,X^C)\ \cong~ &\mathcal{C} (A,X^C) {\texttimes} \mathcal{C} (B,X^C)\ \cong~ &\mathcal{C} (A{\texttimes} C,X) {\texttimes} \mathcal{C} (B{\texttimes} C,X)\ \cong~&\mathcal{C} (A{\texttimes} C+B{\texttimes} C,X)\ \end{aligned}