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The definitions are indeed equivalent. The idea of 'local smallness' is to get for any $X$,$Y$ in $\mathcal{C}^I$ an object of your indexing category to represent, as it were, all (vertical) morphisms between $X$ and $Y$. Both definitions describe this fact, although Johnstone's is, I guess, slightly more 'general' than it needs to be in that it applies the above property to any $X$ and $Y$ in $\mathcal{C}$ (and not to $X$, $Y$ in the same fibre), but that's OK by Theorem 10.1 in Streicher since $\mathcal{S}$ has finite limits. The equivalence of the definitions can also be proved as exercises 8.8.9 and 8.8.10 in Volume 2 of Borceaux.

To see that they are equivalent let for simplicity $X$ and $Y$ lie on the same fibre $I$ (WLOG bearing in mind what I said above.) Then Johnstone's definition says that there exists an arrow $\alpha \colon J \rightarrow I$ and a morphism $f \colon \alpha^*X \rightarrow \alpha^*Y$ such that for any $\beta \colon K \rightarrow I$ and any morphism $g \colon \beta^*X \rightarrow \beta^Y$ beta^*Y$ there exists a unique $u: K \rightarrow J$ such that

$u^(f) $u^{*}(f) = g$ g$$

and $\alpha \circle circ u = \beta$. But now if you write $J$ as $H_{X,Y}$ and $\alpha$ as $h_{X,Y}$ you'll see that the last sentence says exactly that there is a bijection between morphisms $f \colon \beta^*X \rightarrow \beta^*Y$ and morphisms $u$ such that $h_{X,Y} \circle circ u = \beta$, i.e. between morphisms from $\beta$ to $h_{X,Y}$ in $\mathcal{S}/I$. And this is exactly the second definition.
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The definitions are indeed equivalent. The idea of 'local smallness' is to get for any $X$,$Y$ in $\mathcal{C}^I$ an object of your indexing category to represent, as it were, all (vertical) morphisms between $X$ and $Y$. Both definitions describe this fact, although Johnstone's is, I guess, slightly more 'general' than it needs to be in that it applies the above property to any $X$ and $Y$ in $\mathcal{C}$ (and not to $X$, $Y$ in the same fibre), but that's OK by Theorem 10.1 in Streicher since $\mathcal{S}$ has finite limits. The equivalence of the definitions can also be proved as exercises 8.8.9 and 8.8.10 in Volume 2 of Borceaux.

To see that they are equivalent let for simplicity $X$ and $Y$ lie on the same fibre $I$ (WLOG bearing in mind what I said above.) Then Johnstone's definition says that there exists an arrow $\alpha \colon J \rightarrow I$ and a morphism $f \colon \alpha^*X \rightarrow \alpha^*Y$ such that for any $\beta \colon K \rightarrow I$ and any morphism $g \colon \beta^*X \rightarrow \beta^Y Y$ there exists a unique $u: K \rightarrow J$ such that $u^(f) = g$ and $\alpha \circle u = \beta$. But now if you write $J$ as $H_{X,Y}$ and $\alpha$ as $h_{X,Y}$ you'll see that the last sentence says exactly that there is a bijection between morphisms $f \colon \beta^*X \rightarrow \beta^*Y$ and morphisms $u$ such that $h_{X,Y} \circle u = \beta$, i.e. between morphisms from $\beta$ to $h_{X,Y}$ in $\mathcal{S}/I$. And this is exactly the second definition.
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The definitions are indeed equivalent. The idea of 'local smallness' is to get for any $X$,$Y$ in $\mathcal{C}^I$ an object of your indexing category to represent, as it were, all (vertical) morphisms between $X$ and $Y$. Both definitions describe this fact, although Johnstone's is, I guess, slightly more 'general' than it needs to be in that it applies the above property to any $X$ and $Y$ in $\mathcal{C}$ (and not to $X$, $Y$ in the same fibre), but that's OK by Theorem 10.1 in Streicher since $\mathcal{S}$ has finite limits. The equivalence of the definitions can also be proved as exercises 8.8.9 and 8.8.10 in Volume 2 of Borceaux.

To see that they are equivalent let for simplicity $X$ and $Y$ lie on the same fibre $I$ (WLOG bearing in mind what I said above.) Then Johnstone's definition says that there exists an arrow $\alpha \colon J \rightarrow I$ and a morphism $f \colon \alpha^*X \rightarrow \alpha^*Y$ such that for any $\beta \colon K \rightarrow I$ and any morphism $g \colon \beta^*X \rightarrow \beta^Y there exists a unique $u: K \rightarrow J$ such that $u^(f) = g$ and $\alpha \circle u = \beta$. But now if you write $J$ as $H_{X,Y}$ and $\alpha$ as $h_{X,Y}$ you'll see that the last sentence says exactly that there is a bijection between morphisms $f \colon \beta^*X \rightarrow \beta^*Y$ and morphisms $u$ such that $h_{X,Y} \circle u = \beta$, i.e. between morphisms from $\beta$ to $h_{X,Y}$ in $\mathcal{S}/I$. And this is exactly the second definition.