Output value based on a match from different columns

Question

I have two DataFrames df and df1. Thereby df gets daily new inputs via the cat_Id and gives the quality value. Df1 corresponds to a "catalog" that lists all Ids and outputs the name of the corresponding Id.

cat_id= [2,1,3]
begin_date = '2010-10-01'
quality = [35.9,69.1,0.0]
df = pd.DataFrame({'date':pd.date_range(begin_date, periods=len(cat_id)),'cat_id':cat_id,'quality':quality,})

AND: 

category = ['A','B','C']
cat_id= [1,2,3]
df1 = pd.DataFrame({'category':category,'cat_id':cat_id})

So the Output looks like this:

df 
    date     cat_id quality
0   2010-10-01  2   35.9
1   2010-10-02  1   69.1
2   2010-10-03  3   0.0


df1
    category    cat_id
0   A             1
1   B             2
2   C             3

I want to build a function that takes the value of each row of df['cat_id'] and goes through the catalog df1[cat_id] and then takes the associated name from df1[category].

So basically the Goal-Output should look like this:

    date      category  quality
0   2010-10-01  B       35.9
1   2010-10-02  A       69.1
2   2010-10-03  C       0.0

Thanks for your help.