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reviewed  Reviewed Learning roadmap in Algebra 
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comment 
Three questions on $\operatorname{hocolim}$
(continuation) The preceding counterexample is very similar to the one in the second part of my answer above. However, there is a subobject of the 1truncation of the bar construction which will compute the homotopy colimit of $F$: at level one, consider only the morphisms $n \to n+1$ for $n\in\mathbb{N}$, and remove all other morphisms from the 1truncated bar construction. The realization of this subobject is precisely the mapping telescope of the functor $F$, and is therefore equivalent to the homotopy colimit of $F$. 
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comment 
Three questions on $\operatorname{hocolim}$
Dear @jeffrey, in the case of the category $\newcommand{\NN}{\mathbb{N}} \NN$, we cannot compute the homotopy colimit of a functor $F:\NN\to\mathrm{sSet}$ using the 1truncation of the bar construction $B(F,\NN,\ast)$ (this bar construction is precisely the "$\coprod$thing"). In fact, the realization of that 1truncation for $F=\ast$ is the 1skeleton of the classifying space of $\NN$, which consists of the set $\NN$ with one segment added between each pair of distinct points of $\NN$. This space is not contractible, thus cannot be the homotopy colimit of $F=\ast$. (to be continued...) 
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Can a birational morphism between smooth varieties be dominated by smooth blowup sequences?
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Does this integral have a closed form?
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comment 
How do I calculate adjoint functors to forgetful functors?
Please do not post questions simultaneously to MathOverflow and to Math.StackExchange. It leads to duplication of effort and is frowned upon by both communities. Please pick one of the sites and wait at least a few days for an answer before reposting the question elsewhere. Thank you. That being said, the present question is not researchlevel and is more appropriate on Math.StackExchange. 
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reviewed  Close How do I calculate adjoint functors to forgetful functors? 