left adjoint preserves colimits

1. Proposition

Let 20231014-left_adjoint_preserves_limits_2d5d1c2bfbb1a2fca757f722875cd0787290c048.svg be categories, functors and adjoints with as left adjoint. Then preserves colimits

2. Proof

2.1. construction and uniqueness

Consider following cocone in 20231014-left_adjoint_preserves_limits_760824120891fc757bc0445145b027d09b237baa.svg , where is a colimit.

20231014-left_adjoint_preserves_limits_13dc13dc40bb8c809d4951b36cb05291ae39818b.svg

Applying 20231014-left_adjoint_preserves_limits_cf6bf1aab868c8bfacb8fd9d7c1bd31022589360.svg results in following cocone

20231014-left_adjoint_preserves_limits_21034ce7391b1821af9bd579f23087b115d89b36.svg

Suppose there exists an object 20231014-left_adjoint_preserves_limits_c06b0a5a737e8840792537e3e805d3fddef870fc.svg and morphisms making following diagram commute

20231014-left_adjoint_preserves_limits_b7c02697825aa084d50daeec3d5bac9094a6250e.svg

Then we have to show, that there exists a unique morphism 20231014-left_adjoint_preserves_limits_6e534dc2d90c34ddf5922e37195ef905cbc4221d.svg making the diagram commute. Applying results in

20231014-left_adjoint_preserves_limits_d62b1712ee574ee752c3005c284733275b695aa7.svg

By adjointness, we get an adjunction

20231014-left_adjoint_preserves_limits_9bdf11bae39652a146699bc3bcb39e5ac9292869.svg

Let 20231014-left_adjoint_preserves_limits_68c6d54b0d5bf34714fb67afe8e7b58708dde14f.svg .

By naturalness of 20231014-left_adjoint_preserves_limits_9485e48332c47acf9b1f77913ef6fd31df3dc164.svg we get following commutative diagrams ¹

20231014-left_adjoint_preserves_limits_da12f4d72060f4403456f7fe4a5f8db81a2b625d.svg

and precomposition

20231014-left_adjoint_preserves_limits_e9502a0a5e2a51149d68c508cc7c6c8d94e9967c.svg

respectively

20231014-left_adjoint_preserves_limits_bd776405630a2ec21a846187eaac506a60be6358.svg

Thus applying to

20231014-left_adjoint_preserves_limits_6b3e23488aa875fb91042ac97f7c006962a68afc.svg

Hence

20231014-left_adjoint_preserves_limits_adfaf6264c8915ba35eca7d86c9c6b241a138f83.svg

is a cocone By universal property of 20231014-left_adjoint_preserves_limits_2410cc5ec8ea70e423495e94d4de1f9fb2faa43e.svg , there exists a unique morphism

20231014-left_adjoint_preserves_limits_8bab8c593205cc2fbfff8ea672981bf69e4c82b6.svg

and by adjointness

20231014-left_adjoint_preserves_limits_cc10c02fb94d067292f92fb3661e2c24a2c82b94.svg

Thus there exists a unique morphism

20231014-left_adjoint_preserves_limits_faed8b83957cd6c994d2a22f138c5605cb8773e9.svg

2.2. commuting

w.l.o.g. for

20231014-left_adjoint_preserves_limits_3ef5a28987de9fdaff889bad5d7d55981292eb19.svg

the other side follows analogue

By assumption

20231014-left_adjoint_preserves_limits_8196f12f1ee127f7016471077f15794f3e20ceed.svg

By adjointness, we get following commuting diagram

20231014-left_adjoint_preserves_limits_d4c3d0e008a39d9b1447a34b7705267a323680e2.svg

and thus

20231014-left_adjoint_preserves_limits_330bc740573353d799c9043a48ae5f80804419bb.svg

Furthermore, since

20231014-left_adjoint_preserves_limits_e3731d526ed786f0cd664fb903d341f74d729bd7.svg

we get

20231014-left_adjoint_preserves_limits_d3bf4fbae5783fca23f0980fafa431f636ce2ad9.svg

Hence we finally get

20231014-left_adjoint_preserves_limits_669a51a31206ac4ca221a5eeba2aafe3676fbb8d.svg

3. Alternative short version

Let 20231014-left_adjoint_preserves_limits_527be79e7e81f054b494c0f4e81e5b55519e0310.svg be a colimit. Then we get by definition of a left adjoint

20231014-left_adjoint_preserves_limits_dd3a0a22b74b5e933375e5f80de665c5f12cbd67.svg

Using that contravariant hom-functor dualizes small colimits to limits we get

20231014-left_adjoint_preserves_limits_d992deb2be562de12c818a8f0281e00f8ee58c76.svg

Again applying the left-adjoint and dualizing of colimits gives

20231014-left_adjoint_preserves_limits_aa5ae18565724d6cfccbb67fd7af78e6e6fb1b22.svg

Footnotes:

slight abuse of notation, formally we would compose with 20231014-left_adjoint_preserves_limits_5360d5e6a1bd3328c8d0716f6abd9aa5956b9e8a.svg on the other side.