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The definition of initial and terminal objects given here http://en.wikipedia.org/wiki/Initial_and_terminal_objects makes sense to me.

The definition of initial and terminal morphisms given here http://en.wikipedia.org/wiki/Universal_property#Initial_morphism makes sense to me.

Now, suppose $C$ some category and $U:C\rightarrow C$ is some endofunctor.

What relationship, if any, is there between the initial objects of $C$ and the initial objects of $(X\downarrow U)$ for some object $X$ in $C.$

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    $\begingroup$ "An initial morphism from X to U is an initial object in the category (X \downarrow U) of morphisms from X to U" (Wikipedia) $\endgroup$ – Martin Brandenburg Oct 18 '14 at 21:23
  • $\begingroup$ Thanks for the clarification. I didn't have my question quite right. (I think) I fixed it. $\endgroup$ – Randy E Oct 18 '14 at 21:40
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Martin Brandenburg has commented that an initial morphism is an initial object in a certain category. I want to point out that conversely, an initial object can be viewed as a special case of an initial morphism.

Let $D$ be any category. Let $C$ be the terminal category consisting of a single object $X$ and a single arrow $\text{id}_X:X\to X$. Let $U:D\to C$ be the unique functor sending every object to $X$ and every arrow to $\text{id}_X$. Then an initial morphism from $X$ to $U$ is defined by an object in $D$, call it $A$, and a morphism in $C$ from $X$ to $U(A) = X$ (the only choice is $\text{id}_X$).

Now the definition reads that for any object $Y$ in $D$ and any morphism $f:X\to U(Y)$ in $C$, there is a unique morphism $g:A \to Y$ in $D$ such that a diagram commutes: $U(g)\circ \text{id}_X = f$ in $C$. But the choice of the morphism $f$ and the commutativity of the diagram are trivial, so this boils down to the statement that for any $Y$ in $D$, there is a unique morphism $g:A\to Y$, i.e. $A$ is an initial object in $D$.

Another way of saying this is that the functor $A:C \to D$ which sends the unique object in $C$ to the initial object in $D$ is left adjoint to the trivial functor $U:D\to C$. Note that $\text{Hom}_C(X,U(Y))$ consists of exactly one element ($\text{id}_X$) for all $Y$, and $\text{Hom}_D(A(X),Y)$ consists of exactly one element for all $Y$ if and only if $A(X)$ is initial.

Oh, you've just changed your question, so my answer no longer applies. Still, you may find it helpful.

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  • $\begingroup$ Thanks for the comments. I'm still new to category theory, so I'm having some trouble getting my wording right. $\endgroup$ – Randy E Oct 18 '14 at 21:45
  • $\begingroup$ Alex it is a good answer in search of a question :-) in general limits, universals, adjoints and kan extensions can all be expressed in term of each other. This is explained eg. In CWM $\endgroup$ – magma Oct 20 '14 at 13:08

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