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I have to see that every left adjoint functor preserves initial objects.

I prove it by Adjoint functor theorem which states that under certain conditions a functor that preserves colimits is a left adjoint. A basic result of the category theory is that left adjoint preserves all colimits, which can be characterized as initial objects.

Is this idea correct to prove this statement "Every left adjoint funtor preserves initial objects"? or how can we see this.

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Yes. Initial objects are colimits. And left adjoints preserve colimits. Thus left adjoints preserve initial objects. – Joe Johnson 126 Jan 6 '14 at 15:48
up vote 5 down vote accepted

If $F:X\to A$ is left adjoint to $G:A\to X$ and $i$ is initial in $X$, then the hom-set $A(Fi,a)$ is naturally isomorphic to the hom-set $X(i,Ga)$. Since the latter has only one element for every $a\in A$, so does the former, which proves that $Fi$ is initial in $A$.

More generally, a left adjoint functor preserves colimits, and an initial object is simply a colimit over the empty diagram.

When you wrote ".. left adjoint preserves all colimits, which can be characterized as initial objects.", I'm not sure if what you actually meant to say was that "initial objects can be expressed as colimits." However, a colimit is indeed an initial object. Namely, an initial object in the category $$(D\downarrow\Delta X)$$ where $D:J\to X$ is the diagram in $X$ (as an object in $X^J$) and and $\Delta:X\to X^J$ is the diagonal functor sending an object $x$ to the constant diagram $\Delta x:J\to X$ whose only morphism is $1_x$.

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