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In an arbitrary category $\mathcal C$, a subobject of an object $X$ is a monomorphism $m\colon X'\to X$.

The intersection of a class of subobjects $\langle m_i\colon X_i\to X\rangle_{i\in I}$ is defined to be the subobject $m\colon X'\to X$ such that

(1) $m$ factors through each $m_i$,

(2) if a morphism $f\colon Y\to X$ factors through each $m_i$, then it factors through $m$.

Now, the intersection to me seems to be the multiple pullback of the sink $\langle m_i\colon X_i \to X\rangle$, i.e., the limit over it. Indeed, by definition the limit is a morphism $m\colon X'\to X$ such that $m$ factors through each $m_i$ (condition (1) above), and if we are given a cone $\langle f_i\colon Y\to X_i \rangle_{i\in I}$ by composition we have $f\colon Y\to X$ factoring through each $m_i$ which gives us a factorization through $m$ by the universal property. The only thing left to check is $m\colon X'\to X$ is a monomorphism. But this follows again by the universal property of the limit.

My point of confusion is on pg. 211 in The Joy of Cats which gives the definition of a complete category as one which has all small limits, and a strongly complete category as one which is complete and has intersections.

If a category is complete, it has multiple pullbacks and hence, has intersections. So isn't the definition redundant, i.e., complete categories and strongly complete categories are the same? I must be making a mistake in some definition.

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Your observation is correct, intersections are just special cases of limits.

In the definition of an intersection (Def. 11.23 in Joy of Cats) it is not assumed that the indexing set $I$ is small. Therefore if a category has small limits, it doesn't need to have all intersections (only small ones). In well-powered categories this doesn't make any difference (and most categories are well-powered).

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