Theorem. limits in a functor category [leinster2016basic, 6.2.5] [tt-0045]
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Limits in a functor category are computed pointwise.

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This theorem helps proving Lemma [tt-0047].

References

Reference. Basic category theory [leinster2016basic]

2016
Tom Leinster
https://arxiv.org/abs/1612.09375

@article{leinster2016basic,
 title = {Basic category theory},
 author = {Leinster, Tom},
 year = {2016},
 url = {https://arxiv.org/abs/1612.09375},
 journal = {arXiv preprint arXiv:1612.09375}
}

Backlinks

Corollary. evaluation functor preserves limits of ... [leinster2016basic, 6.2.6] [tt-0046]

An important corollary of Theorem [tt-0045].

The Yoneda embedding preserves limits, for any small category [leinster2016basic, 6.2.12]. But it does not preserve colimits in general [leinster2016basic, 6.2.14].

Every presheaf can be expressed as a colimit of representables in a canonical (though not unique) way, this is Theorem [tt-004D], and dual to Lemma [tt-002N].

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Related

Lemma. limits commute with limits [leinster2016basic, 6.2.8] [tt-0047]

Let \({\cal I}\) and \({\cal J}\) be small categories. Let \({\cal C}\) be a locally small category with limits of shape \({\cal I}\) and of shape \({\cal J}\).

Define \[\begin {array}{llll} \mathscr {D}^{\bullet }: & {\cal I} & \to & {[{\cal J}, {\cal C}]} \\ & I & \mapsto & \mathscr {D}(I,-) \end {array}\] and \[\begin {array}{rrrr} \mathscr {D}_{\bullet }: & {\cal J} & \to & {[{\cal I}, {\cal C}]} \\ & J & \mapsto & \mathscr {D}(-, J) \end {array}\]

Then for all \(\mathscr {D}: {\cal I} \times {\cal J} \to {\cal C}\), we have \[ \lim _{\leftarrow {\cal J}} \lim _{\leftarrow {\cal I}} \mathscr {D}^{\bullet } \cong \lim _{\leftarrow {\cal I} {\cal J}} \mathscr {D} \cong \lim _{\leftarrow {\cal I} \leftarrow {\cal J}} \mathscr {D}_{\bullet } \] and all these limits exist. In particular, \({\cal C}\) has limits of shape \({\cal I} \times {\cal J}\).