Are there ever non-universal cones to the identity functor? An initial object in e in C is also a universal cone to the identity functor, but can there ever be a category where an object k in C exists with a cone to the identity functor, but k is not initial. It would seem that in all examples I can think of, a cone to the identity functor must equalize all maps in the category, and so can only come from an initial object.  
 A: Not every cone for the identity functor comes from an initial object. For instance, if $C$ has a zero-object, then for any object $A$ there is an obvious structure of cone for the identity functor given by the family of zero morphisms with domain $A$. 
In general, $A$ admitting a cone for the identity functor amounts to saying that there is a coconstant morphism $A\to B$ for each $B$.
A: Note that if $\gamma: id_C \Rightarrow const_t$ is a cocone, then the component $\gamma_t: t \to t$ is an idempotent. Splitting this idempotent yields a terminal object. By the universal property of the terminal object, cocones on the identity functor are then in bijection with pointed objects of $C$ (a pointed object of $C$ is an object $c \in C$ equipped with a map $1 \to c$). The bijection sends a pointed object $f: 1 \to c$ to $f \circ \gamma$ where $\gamma$ is the canonical cocone from the identity to the terminal object.
In general the idempotent $\gamma_t$ need not split, but one can still say the following. A cocone on $id_C$ exists if and only if the idempotent completion $\tilde C$ of $C$ has a terminal object, and cocones on $id_C$ are in bijection with pointed objects of $\tilde C$ whose underlying $\tilde C$-object lies in $C$.
