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In Adamek and Rosickys' Locally Presentable and Accessible Categories, I came across the following statement (I'm paraphrasing),

every $\mu$-presentable object in a locally $\lambda$-presentable category (for regular cardinals $\mu\geq\lambda$) is a $\mu$-small colimit of $\lambda$- presentable objects. The proof is rather technical, however the following weaker statement is trivial: each $\mu$-presentable object is a split quotient of a $\mu$-small colimit of $\lambda$-presentable objects.

The book then goes on to prove the second statement. But, doesn't this statement directly prove the first as a split quotient can be expressed as a coequaliser between the corresponding split idempotent and the identity map, making it a coequaliser of a $\mu$-small colimit of $\lambda$-presentable objects and hence a $\mu$-small colimit of $\lambda$-presentable objects itself?

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    Suppose $x$ is the colimit at issue and $y$ is its split quotient, the given $\mu$-presentable object. What's your proposed diagram colimiting to $y$ consisting only of $\lambda$-presentable objects? The diagram defining $y$ uses $x$ twice, and only one of those occurrences can be removed via $x$'s universal property. – 2017-01-11
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    Look at Makkai-Paré book, 2.3.11, 2.3.3, 2.1.2. – 2018-01-17

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Suppose $X_j$ is a diagram with shape having $\Omega$ many $\lambda$-presentable objects and $\Omega\leq \kappa<\mu$ many morphisms, and $X_j\xrightarrow{f_j}X$ its colimit. When $\lambda\leq\mu$, the $\lambda$-presentable $X_j$ are also $\mu$-presentable, so $X$ is $\mu$-small colimit of $\mu$-presentable objects, and hence $\mu$-presentable.

As a special case, any (split) quotient $X\rightrightarrows X\to Z$, $Z$ is also $\mu$-presentable since it is the colimit of a finite diagram of $\mu$-presentable objects.

By combining the two diagrams, you get that $Z$ is the colimit of the diagram $X_j\xrightarrow{f_j}X\rightrightarrows X$, whose shape has size at least $2^{\Omega}+\kappa$, which may be larger than $\mu$ if $2^\Omega>\mu$. In particular, the resulting diagramin does not have to be $\mu$-small. For this reason, the converse of the first statement does not follow directly from the converse of the second.

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    Well, and your diagram also fails to consist of $\lambda-$presentable objects, no? – 2017-01-11
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    ... yes, of course. – 2017-01-11
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    Just wondering why to bother with the counting of arrows. – 2017-01-11
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    because sometimes I miss the obvious... – 2017-01-11
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    Ah, sure. I thought you were ...-ing at me, not at yourself. That's a nice argument, anyway, not sure I've ever seen it used outside of proving that a large-complete category is a preorder. – 2017-01-11
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    Wait, so if consider a set of objects $\mathcal{A}$ and look at a $\lambda$ small diagram with each object in the target of the diagram a $\lambda$-small colimit of objects in $\mathcal{A}$ then the resultant colimit may not be a $\lambda$-small colimit over $\mathcal{A}$? – 2017-01-12
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    There are two problems; 1. a $\lambda$-small diagram with each object in the target a $\lambda$-small colimit of objects in $\mathcal A$ is not a $\lambda$ small diagram with each object in $\mathcal A$ (since $\mathcal A$ need not be closed under $\lambda$-small colimits), and 2. if you try to expand the $\lambda$-small diagrams that give the objects in question, the resulting diagram both includes the $\lambda$-small colimits that are not in $\mathcal A$, and is not necessarily $\lambda$-small. – 2017-01-12
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    Look at Makkai-Paré book, 2.3.11, 2.3.3, 2.1.2. – 2018-01-17