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Hello !

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.

Hello !

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i\_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.

Hello !

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.

Hello !

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i\_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i\_*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.

Hello !

$i^*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.

$i\_*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ such that $p \circ f =Id\_F$ this can be express as an equaliser or with internal language as you prefers)

You also have a $i\_!$ functor (left adjoint to $i^*$ ) which is just the forget functor who send $p:Y \rightarrow F$ to $Y$.

For your second question :

You can chose any generating family $B$ of $T$ (for example the image by the yoneda embeddings of a site of definition of $T$) add $F$ to this familly, the familly $B'$ you obtain seen as a full subcategory of $T$ and endowed with the canonical topology of $T$ is (by the Grothendieck comparison lemma) a site of definition of $T$, and you simply choose $U$ to be $F$.