The urge to combine 1- and 2-morphisms in slicing a 2-category. Suppose that $C$ is a 2-category, perhaps $C=\rm{Cat}$, the 2-category of small categories, functors, and natural transformations.  Let $T$ be an object in $C$.  
I form the new 1-category whose objects are morphisms $f\colon A\rightarrow T$ in $C$, and in which a morphism from $f$ to some $f'\colon A'\rightarrow T$ consists of a pair $(\phi,\phi^\sharp)$ where $\phi\colon A\rightarrow A'$ is a 1-morphism in $C$ and $\phi^\sharp\colon\phi\circ f'\rightarrow f$ is a 2-morphism between arrows $A\rightarrow T$. Call this new category the $(C\Uparrow T)$.  An obvious variation comes about by reversing the direction of the 2-morphism, i.e. we could take $\phi^\sharp\colon f\rightarrow\phi\circ f'$; perhaps I might call this variation $(C\Downarrow T)$.
What is the high-brow way to refer to these strange slice-categories?  How do you locate them within a good understanding of 2-categories?  Where are the properties of such things discussed?  What is the relation between these strange slices and the usual 2-categorical slices?
Thanks!
 A: Interesting question!
If C is a 1-category then the overcategory C/T can be described as the lax (or oplax, I forget) limit of the diagram • → C in the 2-category Cat, where the arrow is given by the object T of C.  The lax limit means we ask for a universal limit cone on the diagram where the triangle is filled with a noninvertible 2-morphism.  We can adapt this definition to any object C of any 2-category equipped with a map T from the terminal object.
When C is a 2-category, I believe we need to ask for a "very lax" limit, in which the triangle is not even filled by a (noninvertible) natural transformation, but only a lax (or oplax, depending on which of your constructions you want) natural transformation.  As far as I can see, there is no way to perform your constructions starting only with 2Cat as a 3-category and the data of C and T; we need the extra structure of the (op)lax natural transformations in 2Cat.  Moreover, there is no 3-category of 2-categories, functors, and lax natural transformations in which to take a lax limit and obtain your constructions.  So, these "lax" overcategories are still rather mysterious to me.
I assume by "the usual 2-categorical slices" you mean to require the 2-morphism between $\phi \circ f'$ and $f$ to be invertible, which is the (op?)lax limit of the diagram • → C in 2Cat.
A: For what it's worth, the construction you describe features prominently in https://arxiv.org/abs/0807.4146, though that paper does not use 2-categorical language.
More generally, given a 2-category $C$ (which I usually assume is pivotal, though maybe that's not necessary here), one can construct a 1-category $D$ whose objects are 1-morphisms $f:a\to b$, and whose morphisms are "rectangles": the domain $f:a\to b$ along the bottom, the range $f':a'\to b'$ along the top, additional 1-morphisms (of $C$) $g:a\to a'$ and $h:b\to b'$ along the right and left sides, and a 2-morphism of $C$ filling in the rectangle.  Composition in $D$ is given by stacking the rectangles vertically.  I like to think of the pair $(g, h)$ as the (bi)grading of the morphisms of $D$.  What you describe is the case where we restrict $h$ to be an identity 1-morphism (of $C$).  In the paper linked to above we put an inner product on $D$ and complete it to a von Neumann algebra (in fact, a factor).
In response to David's comment below:
Modulo some details, a planar algebra is equivalent to a pivotal 2-category whose 2-morphisms are vector spaces and whose 1-morphisms are finitely generated.  The standard example is constructed from a pair of factors (irreducible von Neumann algebras) $N\subset M$.  From this data we construct a 2-category whose objects are $N$ and $M$, whose 1-morphisms are generated by the two bimodules $_N M_M$ and $_M M_N$, and whose 2-morphisms are intertwinors.
(So for example the 1-morphisms are
$M\otimes_N M\otimes_N\cdots\otimes_N M$, thought of as either an $N$-$N$ or $N$-$M$ or $M$-$N$ or $M$-$M$ bimodule.)  You can think of the usual planar algebra definition as axiomatizing the "string diagrams" you would draw for this 2-category.
The diagrams in the paper I referred to are rotated 90 degrees from my explanation above.  The left and right sides of the rectangles in the paper correspond to the $f$ and $f'$ of your (David's) original question.  The tops of the rectangles corresponds to your $\phi$, and the interiors of the rectangles correspond to your $\phi^\sharp$.
A: The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$.  The defining universal property is the same as for comma objects, except that the 2-cells in the squares are lax natural transformations.  Your first definition should be the oplax version.
See Kelly, On clubs and doctrines, LNM 420, or Gray, Adjointness For 2-Categories, LNM 391, who calls these '2-comma categories'.
In more detail, Gray's 2-comma categories come from (strict, I think) 2-functors $A \overset{F}{\rightarrow} K \overset{G}{\leftarrow} B$.  An object is a 1-cell $FA \to GB$, a morphism is a square with a 2-cell in, and a 2-cell is given by a pair of 2-cells in $K$ that fit into a commuting cylinder (it's pretty obvious if you draw a picture).  In your example, (what I've called) $C // T$ has 2-cells $(\phi,\phi^\sharp) \Rightarrow (\psi,\psi^\sharp)$ given by 2-cells $\alpha \colon \phi \Rightarrow \psi$ such that $\psi^\sharp \circ f'\alpha = \phi^\sharp$.  (Again, pictures make it much clearer!)  So your slices are actually 2-categories, coming from $C \overset{1}{\rightarrow} C \overset{T}{\leftarrow} \bullet$.  
